Fields Institute Communications Volume 00, 0000 The Cunningham Project Samuel S. Wagstaff, Jr. Center for Education and Research in Information Assurance and Security and Department of Computer Sciences, Purdue University West Lafayette, IN 47907-1398 USA [email protected] Abstract. The goal of the Cunningham Project is to factor numbers of the form bn 1 for small b. We explain why people factor these particular numbers, tell about those who have factored them, list some methods they used and describe some of their outstanding successes. 1 Introduction The goal of the Cunningham Project is to factor numbers of the form bn 1 for integers 2 b 12. The factors of these numbers are important ingredientsin solving many ≤problems≤ in number theory. We will mention some of these problems in the next section. Then we will tell the stories of some of the people who have factored these num- bers over the past two centuries. Naturally, one of them was named Cunningham; we will say a great deal about him. The fourth section explains some of the methods used to factor these numbers. In order to know whether a factorization is complete, we must be able to determine whether a large number is prime or composite. For a long time, primality testing was about as hard as factoring. However, in the past quarter century primality testing has become much easier than factoring. We will discuss the new advances as well as the older methods. Computers and other devices have aided the Cun- ningham Project immeasurably. We will mention some of their achievements and also tell how people factored before computers. In the final section we will mention some of the greatest accomplishments the Cunningham Project has seen. 2 Why are these numbers interesting? In elementary school, we learn how to convert fractions into repeating decimals. For example, 1=37 = 0:027027027027 : : :. The length of the period of the decimal 2000 Mathematics Subject Classification. Primary 11Y05; Secondary 11A41, 11A51, 11Y11. This work was supported in part by grants from the CERIAS Center at Purdue University and from the Lilly Endowment Inc. c 0000 American Mathematical Society 1 2 Samuel S. Wagstaff, Jr. fraction for 1=p, where p is a prime other than 2 or 5, is the smallest positive integer n for which p divides 10n 1. The prime 37 divides 999 but not 99 or 9, so the period of the decimal fraction− for 1=37 is n = 3. The primitive prime factors of 10n 1, that is, the ones which do not divide 10i 1 for any 1 i < n, are the primes− p for which the period of the decimal fraction− 1=p is n. In≤1801 Gauss (see Articles 308{318 of [12]) solved the general problem of determining the period of the decimal fraction for the rational number a=b. Since the ancient Greeks, people have called numbers like 6 and 28, which equal the sum of their proper divisors, \perfect." Euclid knew that if 2p 1 is prime, then 2p−1(2p 1) is perfect. Thus, 22−1(22 1) = 6 and 23−1(23 −1) = 28 are perfect. Euler−proved that all even perfect n−umbers have this form.− The study of perfect numbers led Mersenne to assert which numbers 2p 1 are prime. The p − search for Mersenne primes Mp = 2 1 continues today. For most of the past few hundred years, the largest known prime− has been a Mersenne prime. We still don't know whether there are any odd perfect numbers. Many theorems restrict putative odd perfect numbers in some way. For example, Brent, Cohen and te Riele [5] showed that any odd perfect number must exceed 10300. Furthermore, such a number must have at least 29 prime factors, at least 8 of which are distinct, and one of which exceeds 1; 000; 000. The proofs of these theorems have many cases and require knowledge of factors of numbers of the form bn 1. See Williams [46] for more about the history of perfect numbers and Mersenneprimes. m 2 Fermat thought that Fm = 2 + 1 is prime for every non-negative integer m. The first five of these numbers, F0 = 3, F1 = 5, F2 = 17, F3 = 257 and F4 = 65537, are all prime. If Fermat tried to factor F5 = 4294967297, either he didn't try many divisors or else he made a mistake. Euler showed that F5 = 641 6700417, proving Fermat wrong. Further study of the factorization of numbers 2n·+ 1 suggests that it is most unlikely that there is another prime in the sequence Fm after F4. If N is a large odd number and the factorization of N 1 for Ng+ 1 is known, then it is easy to decide whether N is prime. One can use −a theorem like this one due to Kraitchik [16] and D. H. Lehmer [18] when the factors of N 1 are known. − Theorem 2.1 Let N > 1 and a be integers such that aN−1 1 (mod N). If ≡ a(N−1)=p 1 (mod N) holds for every prime p dividing N 1, then N is prime. 6≡ − The numbers N = bn 1, with small b, are ideal for primality testing via these theorems because we know the prime factors of N 1 = bn. For most b and n it is evident that bn 1 is composite because of an algebraic∓ identity. For example, if n = cd is composite, then bn 1 is divisible by both bc 1 and bd 1. It turns out that when b 2 the numb−ers bn 1 can be prime only− in the −cases 2p 1, m with prime p, and≥b2 + 1. Otherwise,one can exhibit an algebraic factorization− of bn 1. The former numbers are called Mersenne numbers and the latter are generalized Fermat numbers. The factorization of the numbers bn 1 is determined partly by the factorization n of the polynomial x 1. If we let Φd(x) denote the d-th cyclotomic polynomial, then we have − xn 1 = Φ (x) (2.1) − Y d djn for n 1. Although the cyclotomic polynomials are irreducible over the integers, formula≥ (2.1), with x = b, does not give the complete factorization of bn 1 since any − The Cunningham Project 3 factor Φd(b) might be composite. When N = Φd(b) is prime, one can often prove its primality via Theorem 2.1 because the constant term of Φd(x) is 1 for d > 1, and N 1 is frequently divisible by a few powers of b. For example, let b = 12 − 109 and d = 109 (a prime). Then Φ109(x) = (x 1)=(x 1) and N = Φ109(12) = (12109 1)=11 happens to be prime. The number− N 1−= 12(12108 1)=11 is easy to factor− because x108 1 splits into many factors o−ver the integers,− so it is easy to apply Theorem 2.1.−Hugh Williams found this proof in time for the 1983 first edition of [7]. We may factor bn + 1 in a similar fashion using the identity n 2n n x + 1 = (x 1)=(x 1) = Φ (x) Φ (x) = Φ t (x); − − Y d . Y d Y 2 d dj2n djn djm where 2n = 2tm with m odd. Sometimes tables of factored integers can lead us to new algebraic identities. As a simple example, consider this table excerpted from a table of factorizations of numbers 2n + 1. n 2n + 1 2n + 1 factored 2n + 1 factored again 2n=2 + 1 2 5 5 1 5 3 6 65 5 13 5 · 13 9 10 1025 52· 41 25 · 41 33 14 16385 5 29· 113 113 · 145 129 · · · It is easy to observe that the average of the two factors shown in the penultimate column equals the number in the last column. This leads to the identity 24k−2 + 1 = (22k−1 2k + 1)(22k−1 + 2k + 1); (2.2) − which is easy to prove once it is noticed. There is an identity like (2.2) for each b that is not a power. It algebraically factors either bn 1 or bn + 1, depending on b, for all n in a certain arithmetic progression. The tw−o factors are labeled \L" and \M" in [9] and [7]. These identities are named after Aurifeuille, who discovered some of them. (See page v of [9].) In terms of the binary representation of integers, Equation (2.2) shows that there exist integers with arbitrarily many 1 bits which can be multiplied times a number with exactly three 1 bits to give a product with exactly two 1 bits. A polynomial t(x) of degree n > 1 in F2[x] is called primitive if it is irreducible and it does not divide xd+1 for any divisor d of 2n 1. See Golomb [15] for important applications of primitive polynomials in cryptograph− y. The nondivisibility condition in the definition is not hard to check, even when n is in the thousands, if one knows the complete prime factorization of 2n 1. The numbers bn 1 are among the −most interesting large numbers and provide exciting test cases fornew factoring algorithms. 3 The people who have factored the numbers The Cunningham Project takes its name from the book [9] of Cunningham and Woodall. Allan Joseph Champneys Cunningham was born in 1842 in Delhi [43] and edu- cated at King's College, London and at the Military Seminary in Addiscombe. As a military engineer he saw action in Bhutan in 1865{66. The British annexation of Assam State in eastern India in 1826 heightened border tensions with Bhutan.